A053143 -id:A053143 - cp's OEIS frontend

A365345 The number of divisors of the smallest square divisible by n.

1, 3, 3, 3, 3, 9, 3, 5, 3, 9, 3, 9, 3, 9, 9, 5, 3, 9, 3, 9, 9, 9, 3, 15, 3, 9, 5, 9, 3, 27, 3, 7, 9, 9, 9, 9, 3, 9, 9, 15, 3, 27, 3, 9, 9, 9, 3, 15, 3, 9, 9, 9, 3, 15, 9, 15, 9, 9, 3, 27, 3, 9, 9, 7, 9, 27, 3, 9, 9, 27, 3, 15, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3

Offset: 1

Amiram Eldar, Sep 02 2023

The sum of these divisors is A365346(n).

The number of divisors of the square root of the smallest square divisible by n is A322483(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A053143, A322483, A365346, A365489, A365492.

f[p_, e_] := e + 1 + Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> x + 1 + x%2, factor(n)[, 2]));

a(n) = numdiv(n*core(n)); \\ Michel Marcus, Sep 02 2023

a(n) = A000005(A053143(n)).
Multiplicative with a(p^e) = e + 1 + (e mod 2).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 2/p^s - 1/p^(2*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s)^3 * zeta(2*s) * Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n)^2/2 + (3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)) * log(n) + 1 - 3*gamma + 3*gamma^2 - 3*sg1 + (3*gamma - 1)*12*zeta'(2)/Pi^2 + 12*zeta''(2)/Pi^2 + (12*zeta'(2)/Pi^2 + 3*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where
f(1) = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(1) = f(1) * Sum_{primes p} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = 0.7343690473711153863995729489689746152413988981744946512300478410459132782...
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} 4*p*(-1 + 2*p + p^2 - 4*p^3) * log(p)^2 / (1 - 3*p + p^2 + p^3)^2 = 0.1829055032494906699795154632343894745397324334876662084674149254022564139...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A365479 The sum of unitary divisors of the smallest square divisible by n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 17, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 170, 26, 850, 82, 250, 842, 1300, 962, 65, 1220, 1450, 1300, 50, 1370, 1810, 1700, 442, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 410

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest square divisible by n is the same as the number of unitary divisors of n, A034444(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A034444, A034448, A053143, A365346, A365480, A365481.

f[p_, e_] := 1 + p^(e + Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^(f[i,2] + f[i,2]%2) + 1);}

from math import prod
from sympy import factorint
def A365479(n): return prod(p**(e+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A053143(n)).
Multiplicative with a(p^e) = p^(e + (e mod 2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-2) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 - 1/p^4 + 1/p^5 - 1/p^6) = 0.248414056414... .

A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Apr 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A372328.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.
Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .

f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(e=logint(n + 1, 2)); if(n + 1 == 2^e, n, 2^(e+1) - 1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^s(f[i, 2]))};

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - 1).
a(n) = n * A372328(n).
a(n) = n if and only if n is in A036537.
a(n) <= n^2, with equality if and only if n = 1.

A055491 Smallest square divisible by n divided by largest square which divides n.

Original entry on oeis.org

1, 4, 9, 1, 25, 36, 49, 4, 1, 100, 121, 9, 169, 196, 225, 1, 289, 4, 361, 25, 441, 484, 529, 36, 1, 676, 9, 49, 841, 900, 961, 4, 1089, 1156, 1225, 1, 1369, 1444, 1521, 100, 1681, 1764, 1849, 121, 25, 2116, 2209, 9, 1, 4, 2601, 169, 2809, 36, 3025, 196, 3249, 3364

Offset: 1

Henry Bottomley, Jun 28 2000

			a(12) = 36/4 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A056551, A056552.
Cf. A000188, A000290, A007913, A008833, A019554, A053143.
Cf. A013661, A013664.

a055491 = (^ 2) . a007913  -- Reinhard Zumkeller, Jul 23 2014

With[{sqs=Range[100]^2},Table[SelectFirst[sqs,Divisible[#,n]&]/ SelectFirst[ Reverse[sqs],Divisible[n,#]&],{n,60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 18 2018 *)
f[p_, e_] := p^(2 * Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*(f[i,2]%2)));} \\ Amiram Eldar, Oct 27 2022

If n is written as Product(Pj^Ej) then a(n) = Product(Pj^(2*(Ej mod 2))).
a(n) = A053143(n)/A008833(n) = A007913(n)^2 = (A019554(n)/A000188(n))^2 = A000290(n)/A008833(n)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(2))) = 2*Pi^4/945 = 0.206156... . - Amiram Eldar, Oct 27 2022
Dirichlet g.f.: zeta(s-2) * zeta(2*s) / zeta(2*s-4). - Amiram Eldar, Sep 16 2023

A145109 a(n) = 2n core(2*n).

Original entry on oeis.org

4, 4, 36, 16, 100, 36, 196, 16, 36, 100, 484, 144, 676, 196, 900, 64, 1156, 36, 1444, 400, 1764, 484, 2116, 144, 100, 676, 324, 784, 3364, 900, 3844, 64, 4356, 1156, 4900, 144, 5476, 1444, 6084, 400, 6724, 1764, 7396, 1936, 900, 2116, 8836, 576, 196, 100, 10404

Offset: 1

Reikku Kulon, Oct 02 2008

All terms are even squares (A016742).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007913, A016742, A053143.

a[1] = 4; a[n_] := 2*n * Times @@ (First[#]^Mod[Last[#], 2] & /@ FactorInteger[2*n]); Array[a, 50] (* Amiram Eldar, Sep 05 2020 *)

a(n) = 2*n * core(2*n); \\ Michel Marcus, Aug 09 2013

a(n) = A053143(2*n). - Michel Marcus, Aug 09 2013

Definition corrected by Michel Marcus, Aug 09 2013

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A365345 The number of divisors of the smallest square divisible by n.

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A365479 The sum of unitary divisors of the smallest square divisible by n.

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A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

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A055491 Smallest square divisible by n divided by largest square which divides n.

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A145109 a(n) = 2*n * core(2*n).

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A145109 a(n) = 2n core(2*n).